Optimized aerodynamic profile for an arm of a structural casing of a turbine, and structural casing having such an arm

ABSTRACT

When cold and in the non-coated state, the aerodynamic profile is substantially identical to a nominal profile determined by the Cartesian coordinates X,Y, Zadim given in Table 1, in which the coordinate Zadim is the quotient D/H where D is the distance of the point under consideration from a first reference plane P0 situated at the base of the nominal profile, and H is the height of said profile measured from the first reference plane to a second reference plane P1. The measurements D and H are taken radially relative to the axis of the turbine, while the X coordinate is measured in the axial direction of the turbine.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims the benefit of priority to U.S. PatentApplication No. 62/504,567, filed on May 11, 2017, the entirety of whichis incorporated by reference herein.

INTRODUCTION

The present disclosure relates to an aerodynamic profile for an arm of astructural casing of a turbine.

In particular, the disclosure relates to the aerodynamic profile of anarm connecting a central hub and a shroud of such casing.

The disclosure relates in particular to the aerodynamic profile of anarm belonging to a series of arms connecting the central hub and theshroud.

More particularly, the turbine is a low pressure turbine of the typeused in a turbojet of an aircraft.

In particular, the structural casing is an exhaust casing of theturbine.

The exhaust casing may be of the type disclosed in the published USpatent application No. 2007068136, the contents of which is hereinincorporated.

An arm of a structural casing of a turbine is subject to a series ofconstraints. In particular, the aerodynamic profile of such an armparticipates to the global efficiency of the turbine and, as such, itshould enable the turbine to provide the desired efficiency. In order todo that it must be such that the flow of air around the profile issound, i.e. substantially such that it does not give rise to turbulence,which is harmful for overall efficiency.

It must be capable of being installed properly in the environment of theturbine.

Furthermore, the arm must present a profile enabling it to bemanufactured in a reliable and cost-effective manner by availablemanufacturing methods, such as casting, forging, machining, additivefabrication, or else weaving, without this list being limitative.

Furthermore, the profile of the arm must enable it to withstand themechanical stresses to which the arm is subjected, by enabling thosestresses to be spread over the entire arm in such a manner as to avoidpremature wear thereof, both in static and in dynamic modes.

In particular, the arm being a connection arm between a central hub anda shroud of the turbine, it shall be capable of efficiently contributingto ensure proper mechanical connection between the central hub and theshroud, and of being fastened easily to the central hub and to theshroud of the casing. Also, in such case, the arm contributes to theproper centering and positioning of the shroud with respect to thecentral hub.

In addition, in such case, the arm contributes to redressing the flowcoming from the primary thrust nozzle of the turbine engine. Optionally,the arm may serve as a casing for one or more ducts.

An object of the disclosure is to propose an aerodynamic profile for anarm of a structural casing of a turbine having a central hub and ashroud, said profile being optimized, and capable of satisfying thoseobjectives.

This object is achieved by the fact that when cold and in a non-coatedstate, said profile is substantially identical to a nominal profiledetermined by the Cartesian coordinates X,Y,Zadim given in Table 1, inwhich the coordinate Zadim is the quotient D/H where D is the distanceof the point under consideration from a first reference X,Y planesituated at the base of the nominal profile, and H is the height of saidprofile measured from said first reference plane that is theintersection of the stacking axis of the arm and the axisymmetricsurface of the hub, out to a second reference plane that is theintersection of said stacking axis with the axisymmetric inner surfaceof the shroud casing, the measurements D and H being taken radiallyrelative to the axis of the turbine, while the coordinate X is measuredin the axial direction of the turbine.

This profile has been determined as a result of numerous tests andsimulations. It is defined cold, i.e. at an ambient temperature of 20°C. That is a reference temperature at which the profile is geometricallydetermined. The above-mentioned objectives of aerodynamics andmechanical optimization are naturally valid for the conditions of use ofthis aerodynamic profile, i.e. when hot, at a temperature that isstabilized when the engine of which the turbine forms a part is beingused under cruising conditions.

Furthermore, the aerodynamic profile of the disclosure is defined in thenon-coated state. Since arms of turbine casings are subjected to hightemperature gradients, it is common practice for them to be providedwith a coating having thermal properties enabling them more easily towithstand such temperature variations. The profile is determined priorto installing such a coating.

It is stated above that the profile of the disclosure is “substantiallyidentical” to the nominal profile. This means that the profile maydepart very slightly from said nominal profile.

The aerodynamic profile is thus preferably defined within an envelope of±1 millimeter (mm) in a direction normal to the surface of the nominalprofile.

It is also preferable for the X,Y coordinates of the aerodynamic profileto lie within a range of ±5% relative to the X,Y coordinates of thenominal profile.

The aerodynamic profile may be positioned rotated within ±10°,particularly within ±5°, more particularly within ±3°, with respect tothe stacking axis. These variations take account in particular of atleast one of manufacturing tolerances of the profile, the setting of theprofile to adapt to the flow coming from the turbine sections situatedupstream, and possible deformation occurring when mounting the arm andintegrating the same in the turbine.

BRIEF DESCRIPTION OF THE DRAWINGS

The disclosure can be better understood and its advantages appear betteron reading the following detailed description of an embodiment given byway of non-limiting example. The description refers to the accompanyingdrawings, in which:

FIG. 1 is a fragmentary axial section view of a turbine including anexhaust casing having an arm of profile that corresponds to that of thedisclosure;

FIG. 2 is a partial view of the exhaust casing taken on line II-II ofFIG. 1.

DETAILED DESCRIPTION

The turbine shown in axial section in FIG. 1 comprises five stagesreferenced respectively S1 to S5. In the direction DF going fromupstream to downstream, each stage comprises a nozzle that forms aportion of the stator of the turbine and that has a plurality ofradially-oriented vanes, and a rotary wheel that forms a portion of therotor of the turbine and that likewise includes a plurality ofradially-oriented blades.

In FIG. 1, the vanes of the nozzles of stages S1 to S7 are givenreferences N1 to N5, whereas the blades of rotary wheels of stages S1 toS5 are given respective references W1 to W5.

In known manner, the vanes of the nozzles are fastened at both ends tostationary structure portions, while the blades of the rotary wheels arefastened to a rotary disk via their roots that are formed at theirradially-inner ends closer to the axis of the rotation A of the turbine,the disks of the respective wheels being constrained to rotate together.In the example shown, the tips of the blades of the rotary wheels,formed at their radially-outer ends, carry top platforms.

As shown in FIG. 1, the turbine further comprises an exhaust casing 10comprising a central hub 12, a shroud 14 and a plurality of arms 16connecting the hub 12 and the shroud 14. The central hub, the shroud andthe external ring are arranged concentrically.

As mentioned, the exhaust casing may be of the type disclosed by thepublished US patent application No. 2007068136. The arms 16 serve toconnect the central hub 12 to the shroud 14. In a known manner, thecentral hub serves as a bearing support, for bearings supporting rotaryparts of the turbine, especially a rotating shaft (not shown) to whichthe rotary disks are mounted.

As best seen in FIG. 2, and as mentioned in US 2007068136, the arms 16comprise two categories of arms, namely at least one main arm 16A and atleast one secondary arm 16B. The casing may comprise a total of 10 to 20arms, including between 1 and 3 main arms and between 7 and 19 secondaryarms. In particular, the casing may comprise 15 arms including 1 mainarm and 14 secondary arms.

Seen in the circumferential direction, the main arms are thicker thanthe secondary arms. In particular, the main arm may serve as casings forone or more ducts.

The arms 16A and 16B are each attached to the central hub 12 are theirradially inner ends, and are each attached the shroud 14 at theirradially outer ends. The arms and at least part of the central huband/or the shroud may be integrally formed. The arms may be formedintegrally with one or both of the central hub and the shroud. The armsmay be distinct from at least one of the elements comprising the centralhub and the shroud and be welded or affixed in any suitable manner tosaid at least one element.

By convention, the aerodynamic profile of the arm is the entire portionof said arm that extends radially from the external cylindrical surfaceof the central hub to the internal cylindrical surface of the shroud.

The disclosure relates in particular to a main arm 16A connecting thecentral hub and the shroud.

In FIG. 2, there can be seen a frame of reference comprising Cartesiancoordinates X,Y,Z for each one of the arms 16A and 16B. The radialdirection Z is the height direction of the arm, which extends radiallyfrom its root platform to its top platform. This direction Z isperpendicular to the axial direction X which is the direction of theaxis of rotation A of the turbine. The direction Y is perpendicular tothe X,Z plane and is therefore tangential to the direction of rotationof the turbine.

The nominal profile from which the aerodynamic profile of the disclosureis determined is defined in following Table 1 of coordinates, in whichthe coordinate Zadim, measured along the axis Z is non-dimensional,whereas the dimensions X and Y, respectively measured along the axes Xand Y, are expressed in millimeters.

TABLE 1 X Y Zadim −56.9443 −37.9659 −0.2 −57.0582 −38.1951 −0.2 −57.2146−38.5325 −0.2 −57.4342 −39.0628 −0.2 −57.6945 −39.819 −0.2 −57.9503−40.8317 −0.2 −58.1314 −42.1229 −0.2 −58.1465 −43.6967 −0.2 −57.8916−45.5294 −0.2 −57.2688 −47.5662 −0.2 −56.2035 −49.7266 −0.2 −54.6546−51.9183 −0.2 −52.6151 −54.0525 −0.2 −50.1031 −56.052 −0.2 −47.1559−57.8583 −0.2 −43.8212 −59.4339 −0.2 −40.1503 −60.7594 −0.2 −36.1932−61.8276 −0.2 −31.9973 −62.6407 −0.2 −27.6067 −63.208 −0.2 −23.0622−63.5437 −0.2 −18.4015 −63.6648 −0.2 −13.6598 −63.5899 −0.2 −8.86993−63.3396 −0.2 −4.06279 −62.9364 −0.2 0.732695 −62.4028 −0.2 5.489262−61.7619 −0.2 10.18082 −61.0343 −0.2 14.78236 −60.2386 −0.2 19.27067−59.396 −0.2 23.62381 −58.5257 −0.2 27.82082 −57.6445 −0.2 31.84173−56.7657 −0.2 35.66856 −55.9038 −0.2 39.28484 −55.0713 −0.2 42.67563−54.2779 −0.2 45.82774 −53.5312 −0.2 48.73037 −52.8382 −0.2 51.37497−52.2038 −0.2 53.75552 −51.6316 −0.2 55.86881 −51.1234 −0.2 57.71469−50.6798 −0.2 59.29612 −50.2995 −0.2 60.52248 −49.7081 −0.2 61.24685−48.8704 −0.2 61.60194 −48.0781 −0.2 61.72882 −47.4421 −0.2 61.74472−46.9897 −0.2 61.72303 −46.7048 −0.2 61.69125 −46.4877 −0.2 61.65026−46.2961 −0.2 61.58147 −46.0503 −0.2 61.43405 −45.6737 −0.2 61.14592−45.171 −0.2 60.63181 −44.5901 −0.2 59.80325 −44.0495 −0.2 58.61449−43.7918 −0.2 57.16377 −43.875 −0.2 55.471 −43.9803 −0.2 53.5327−44.0971 −0.2 51.349 −44.223 −0.2 48.92288 −44.3543 −0.2 46.26007−44.486 −0.2 43.36884 −44.6123 −0.2 40.25983 −44.7265 −0.2 36.94589−44.82 −0.2 33.44197 −44.8825 −0.2 29.76502 −44.9054 −0.2 25.93379−44.8792 −0.2 21.96874 −44.7926 −0.2 17.8919 −44.6357 −0.2 13.72651−44.4016 −0.2 9.496988 −44.0832 −0.2 5.228591 −43.6751 −0.2 0.946998−43.1753 −0.2 −3.32205 −42.5846 −0.2 −7.55301 −41.9061 −0.2 −11.721−41.146 −0.2 −15.8022 −40.3137 −0.2 −19.7743 −39.4213 −0.2 −23.6167−38.4836 −0.2 −27.3109 −37.5184 −0.2 −30.8412 −36.546 −0.2 −34.1957−35.5946 −0.2 −37.3647 −34.6922 −0.2 −40.3438 −33.8784 −0.2 −43.1318−33.2048 −0.2 −45.7264 −32.7324 −0.2 −48.1166 −32.5425 −0.2 −50.2632−32.6942 −0.2 −52.0973 −33.201 −0.2 −53.5612 −33.9688 −0.2 −54.663−34.8423 −0.2 −55.4621 −35.6899 −0.2 −56.0279 −36.4319 −0.2 −56.4196−37.0295 −0.2 −56.6794 −37.4716 −0.2 −58.2807 −31.7724 −0.1 −58.4027−31.9973 −0.1 −58.5712 −32.3288 −0.1 −58.8098 −32.8509 −0.1 −59.0973−33.5972 −0.1 −59.3895 −34.5999 −0.1 −59.6171 −35.8838 −0.1 −59.6891−37.456 −0.1 −59.5005 −39.2967 −0.1 −58.9517 −41.3547 −0.1 −57.9651−43.5521 −0.1 −56.4964 −45.7984 −0.1 −54.5353 −48.0049 −0.1 −52.0972−50.0937 −0.1 −49.2172 −52.0054 −0.1 −45.9415 −53.7004 −0.1 −42.3209−55.1576 −0.1 −38.405 −56.368 −0.1 −34.2412 −57.3322 −0.1 −29.8739−58.0577 −0.1 −25.3445 −58.5573 −0.1 −20.6912 −58.8467 −0.1 −15.9499−58.9431 −0.1 −11.1541 −58.866 −0.1 −6.33558 −58.6366 −0.1 −1.52395−58.2766 −0.1 3.25266 −57.808 −0.1 7.967434 −57.2503 −0.1 12.59471−56.6214 −0.1 17.11054 −55.9414 −0.1 21.49226 −55.2289 −0.1 25.71837−54.4999 −0.1 29.76839 −53.7669 −0.1 33.62386 −53.0438 −0.1 37.26786−52.3424 −0.1 40.68509 −51.672 −0.1 43.86211 −51.0397 −0.1 46.78788−50.4519 −0.1 49.45366 −49.9135 −0.1 51.85333 −49.4277 −0.1 53.9836−48.9961 −0.1 55.84429 −48.6195 −0.1 57.43843 −48.2966 −0.1 58.68535−47.7498 −0.1 59.43951 −46.9389 −0.1 59.82299 −46.1599 −0.1 59.97277−45.5289 −0.1 60.00499 −45.0773 −0.1 59.9936 −44.7919 −0.1 59.96968−44.5738 −0.1 59.93565 −44.3808 −0.1 59.87577 −44.1327 −0.1 59.74205−43.751 −0.1 59.47226 −43.2383 −0.1 58.97948 −42.6392 −0.1 58.17098−42.069 −0.1 56.99231 −41.7685 −0.1 55.53953 −41.7993 −0.1 53.84407−41.8433 −0.1 51.90281 −41.89 −0.1 49.71598 −41.937 −0.1 47.28671−41.9806 −0.1 44.62088 −42.0159 −0.1 41.72697 −42.0378 −0.1 38.61587−42.0396 −0.1 35.30072 −42.0133 −0.1 31.79682 −41.9493 −0.1 28.12144−41.8394 −0.1 24.29366 −41.6747 −0.1 20.33433 −41.4449 −0.1 16.26581−41.1409 −0.1 12.11159 −40.7565 −0.1 7.896335 −40.2855 −0.1 3.645463−39.7236 −0.1 −0.61528 −39.0694 −0.1 −4.86021 −38.3249 −0.1 −9.0639−37.4941 −0.1 −13.2017 −36.5839 −0.1 −17.2502 −35.6047 −0.1 −21.1874−34.5694 −0.1 −24.9934 −33.4935 −0.1 −28.6504 −32.3955 −0.1 −32.1433−31.2962 −0.1 −35.4612 −30.2243 −0.1 −38.5956 −29.208 −0.1 −41.5433−28.2871 −0.1 −44.3052 −27.5133 −0.1 −46.8811 −26.9474 −0.1 −49.2628−26.6713 −0.1 −51.4135 −26.7454 −0.1 −53.2646 −27.1856 −0.1 −54.7554−27.9001 −0.1 −55.888 −28.7332 −0.1 −56.7172 −29.5514 −0.1 −57.3095−30.2725 −0.1 −57.7224 −30.8555 −0.1 −57.9981 −31.288 −0.1 −58.3262−27.5689 0 −58.4486 −27.7936 0 −58.6176 −28.1248 0 −58.857 −28.6466 0−59.1457 −29.3924 0 −59.4395 −30.3947 0 −59.6691 −31.6782 0 −59.7435−33.2503 0 −59.5578 −35.0913 0 −59.0123 −37.1501 0 −58.0292 −39.3491 0−56.564 −41.5977 0 −54.6063 −43.8072 0 −52.1715 −45.8999 0 −49.2945−47.8161 0 −46.0215 −49.5163 0 −42.4032 −50.9792 0 −38.4892 −52.1957 0−34.3269 −53.1664 0 −29.9608 −53.8988 0 −25.4321 −54.4055 0 −20.7793−54.7022 0 −16.0381 −54.806 0 −11.2423 −54.7365 0 −6.42337 −54.5147 0−1.61118 −54.1623 0 3.166159 −53.7011 0 7.881804 −53.1508 0 12.51007−52.5292 0 17.02695 −51.8563 0 21.40979 −51.1507 0 25.63704 −50.4283 029.68821 −49.7017 0 33.54481 −48.9846 0 37.1899 −48.289 0 40.60818−47.624 0 43.7862 −46.9966 0 46.71288 −46.4135 0 49.3795 −45.8792 051.77993 −45.3972 0 53.91088 −44.969 0 55.77216 −44.5952 0 57.36681−44.2748 0 58.61458 −43.73 0 59.37 −42.9203 0 59.75472 −42.1419 059.90548 −41.5112 0 59.93842 −41.0596 0 59.92747 −40.7742 0 59.90389−40.5561 0 59.87016 −40.363 0 59.81067 −40.1148 0 59.67756 −39.7329 059.40857 −39.2197 0 58.91674 −38.6199 0 58.10913 −38.0484 0 56.93093−37.746 0 55.4781 −37.7745 0 53.78258 −37.8159 0 51.84125 −37.8596 049.65435 −37.9031 0 47.22501 −37.9429 0 44.55913 −37.9741 0 41.66519−37.9913 0 38.55408 −37.9883 0 35.23898 −37.9568 0 31.73519 −37.8873 028.05998 −37.7716 0 24.23247 −37.6009 0 20.2735 −37.3649 0 16.20547−37.0545 0 12.05186 −36.6636 0 7.837349 −36.186 0 3.587366 −35.6173 0−0.67235 −34.9565 0 −4.9161 −34.2053 0 −9.11847 −33.3679 0 −13.2548−32.4512 0 −17.3018 −31.4657 0 −21.2374 −30.4241 0 −25.0417 −29.3423 0−28.697 −28.2385 0 −32.1881 −27.1338 0 −35.5043 −26.0567 0 −38.6371−25.0355 0 −41.5834 −24.1099 0 −44.344 −23.3317 0 −46.919 −22.7619 0−49.3003 −22.482 0 −51.4511 −22.5527 0 −53.303 −22.99 0 −54.7948−23.7022 0 −55.9288 −24.5335 0 −56.7592 −25.3504 0 −57.3526 −26.0705 0−57.7665 −26.6529 0 −58.0429 −27.0849 0 −58.073 −24.0664 0.1 −58.1931−24.2924 0.1 −58.3587 −24.6254 0.1 −58.5926 −25.1496 0.1 −58.8734−25.8984 0.1 −59.1567 −26.9037 0.1 −59.3728 −28.1896 0.1 −59.4308−29.7624 0.1 −59.2259 −31.6014 0.1 −58.6588 −33.6543 0.1 −57.6527−35.8429 0.1 −56.164 −38.076 0.1 −54.1833 −40.2649 0.1 −51.7268 −42.3320.1 −48.8298 −44.218 0.1 −45.5392 −45.8838 0.1 −41.9058 −47.3087 0.1−37.9792 −48.4842 0.1 −33.807 −49.4112 0.1 −29.4335 −50.0979 0.1−24.8998 −50.5571 0.1 −20.2441 −50.8051 0.1 −15.5021 −50.8593 0.1−10.7072 −50.7395 0.1 −5.89091 −50.4673 0.1 −1.08267 −50.0644 0.13.689579 −49.5533 0.1 8.399203 −48.9536 0.1 13.0207 −48.2836 0.117.53029 −47.5634 0.1 21.9055 −46.812 0.1 26.12495 −46.0453 0.1 30.16829−45.2763 0.1 34.01717 −44.519 0.1 37.65478 −43.7852 0.1 41.06591−43.0844 0.1 44.23718 −42.4238 0.1 47.1576 −41.81 0.1 49.81849 −41.24790.1 52.21373 −40.7407 0.1 54.34008 −40.2902 0.1 56.19734 −39.897 0.157.78854 −39.5599 0.1 59.03054 −39.0021 0.1 59.77745 −38.1845 0.1 60.154−37.4021 0.1 60.29814 −36.7698 0.1 60.32634 −36.318 0.1 60.31241−36.0326 0.1 60.28655 −35.8148 0.1 60.2508 −35.6221 0.1 60.18872−35.3746 0.1 60.05162 −34.994 0.1 59.77726 −34.4838 0.1 59.27917−33.8891 0.1 58.46563 −33.3261 0.1 57.28433 −33.0361 0.1 55.83187−33.0798 0.1 54.13688 −33.1389 0.1 52.19611 −33.2029 0.1 50.00978−33.2693 0.1 47.581 −33.3346 0.1 44.91559 −33.3937 0.1 42.02199 −33.44120.1 38.91102 −33.4708 0.1 35.59577 −33.474 0.1 32.09145 −33.4411 0.128.41523 −33.3639 0.1 24.58614 −33.2334 0.1 20.62492 −33.0389 0.116.55385 −32.771 0.1 12.39638 −32.4236 0.1 8.177099 −31.9902 0.13.921394 −31.4661 0.1 −0.34501 −30.8499 0.1 −4.59639 −30.1432 0.1−8.80731 −29.3498 0.1 −12.953 −28.4765 0.1 −17.0101 −27.5334 0.1−20.9564 −26.5332 0.1 −24.7718 −25.4912 0.1 −28.4385 −24.4258 0.1−31.9409 −23.3577 0.1 −35.2683 −22.3153 0.1 −38.4116 −21.327 0.1−41.3674 −20.4323 0.1 −44.136 −19.6831 0.1 −46.7169 −19.1402 0.1 −49.101−18.8853 0.1 −51.2509 −18.9786 0.1 −53.0981 −19.4352 0.1 −54.5824−20.1629 0.1 −55.7076 −21.0061 0.1 −56.5294 −21.8316 0.1 −57.1152−22.5579 0.1 −57.523 −23.1446 0.1 −57.7948 −23.5795 0.1 −57.914 −20.52170.2 −58.0319 −20.7489 0.2 −58.1941 −21.0835 0.2 −58.4229 −21.6099 0.2−58.6964 −22.3614 0.2 −58.9699 −23.3695 0.2 −59.1734 −24.6574 0.2−59.216 −26.2306 0.2 −58.9931 −28.0676 0.2 −58.406 −30.1149 0.2 −57.3785−32.2935 0.2 −55.8681 −34.512 0.2 −53.8661 −36.6814 0.2 −51.3895−38.7244 0.2 −48.4743 −40.5819 0.2 −45.1675 −42.2155 0.2 −41.5203−43.6049 0.2 −37.5825 −44.7419 0.2 −33.4014 −45.6281 0.2 −29.0214−46.272 0.2 −24.4834 −46.6869 0.2 −19.8255 −46.8894 0.2 −15.0832−46.8972 0.2 −10.2898 −46.7306 0.2 −5.47631 −46.4113 0.2 −0.67224−45.9615 0.2 4.094782 −45.4037 0.2 8.798321 −44.7581 0.2 13.41305−44.0429 0.2 17.91539 −43.2786 0.2 22.28304 −42.4845 0.2 26.4948−41.6767 0.2 30.53043 −40.8682 0.2 34.37173 −40.0732 0.2 38.00199−39.3039 0.2 41.4061 −38.5698 0.2 44.57077 −37.8783 0.2 47.48505 −37.2360.2 50.14031 −36.6479 0.2 52.53049 −36.1173 0.2 54.65233 −35.6461 0.256.50566 −35.2347 0.2 58.09349 −34.8821 0.2 59.32998 −34.3122 0.260.06886 −33.4873 0.2 60.43774 −32.7013 0.2 60.5757 −32.0676 0.260.59948 −31.6155 0.2 60.58276 −31.3303 0.2 60.55478 −31.1127 0.260.51714 −30.9204 0.2 60.45264 −30.6735 0.2 60.31183 −30.2943 0.260.03251 −29.7867 0.2 59.52863 −29.1969 0.2 58.70961 −28.6419 0.257.52554 −28.3635 0.2 56.07358 −28.4214 0.2 54.37924 −28.4971 0.252.4392 −28.58 0.2 50.25362 −28.6678 0.2 47.82559 −28.7568 0.2 45.16089−28.8419 0.2 42.26789 −28.9178 0.2 39.15736 −28.9777 0.2 35.8423−29.0133 0.2 32.33782 −29.0147 0.2 28.66102 −28.9735 0.2 24.83084−28.8803 0.2 20.86791 −28.7246 0.2 16.79442 −28.4965 0.2 12.63375−28.1898 0.2 8.410433 −27.7976 0.2 4.149808 −27.3151 0.2 −0.12241−26.7406 0.2 −4.3805 −26.0755 0.2 −8.59897 −25.3233 0.2 −12.753 −24.49060.2 −16.8191 −23.5872 0.2 −20.775 −22.6256 0.2 −24.6004 −21.621 0.2−28.2773 −20.5914 0.2 −31.7901 −19.5576 0.2 −35.1274 −18.5478 0.2−38.2802 −17.5902 0.2 −41.2446 −16.7245 0.2 −44.0205 −16.0024 0.2−46.6065 −15.4847 0.2 −48.993 −15.2532 0.2 −51.1419 −15.3674 0.2−52.9845 −15.8421 0.2 −54.4616 −16.5843 0.2 −55.5785 −17.4384 0.2−56.3923 −18.2719 0.2 −56.971 −19.0039 0.2 −57.373 −19.5946 0.2 −57.6405−20.0321 0.2 −57.8149 −17.0271 0.3 −57.9304 −17.2554 0.3 −58.0891−17.5917 0.3 −58.3122 −18.1206 0.3 −58.5777 −18.875 0.3 −58.8404−19.8859 0.3 −59.0302 −21.1759 0.3 −59.0561 −22.7495 0.3 −58.8137−24.584 0.3 −58.2048 −26.6249 0.3 −57.1542 −28.7925 0.3 −55.6202−30.9947 0.3 −53.5953 −33.1427 0.3 −51.097 −35.1592 0.3 −48.1622−36.9856 0.3 −44.8382 −38.5839 0.3 −41.1764 −39.9343 0.3 −37.2267−41.0294 0.3 −33.0365 −41.8711 0.3 −28.6498 −42.4683 0.3 −24.1077−42.8349 0.3 −19.4479 −42.9877 0.3 −14.7058 −42.945 0.3 −9.91437−42.7274 0.3 −5.1046 −42.3569 0.3 −0.30559 −41.8559 0.3 4.455229−41.2475 0.3 9.151627 −40.5518 0.3 13.75848 −39.7875 0.3 18.25243−38.9754 0.3 22.61138 −38.1348 0.3 26.8143 −37.2822 0.3 30.84109−36.4307 0.3 34.6737 −35.5949 0.3 38.29557 −34.787 0.3 41.69168 −34.01670.3 44.8488 −33.2915 0.3 47.75608 −32.6183 0.3 50.40493 −32.0019 0.352.78933 −31.4459 0.3 54.90603 −30.9521 0.3 56.75488 −30.5211 0.358.33886 −30.1516 0.3 59.56922 −29.5685 0.3 60.29927 −28.7358 0.360.65976 −27.9459 0.3 60.79097 −27.3108 0.3 60.80994 −26.8585 0.360.79018 −26.5735 0.3 60.75988 −26.3562 0.3 60.7202 −26.1643 0.360.65308 −25.918 0.3 60.50823 −25.5404 0.3 60.22352 −25.0358 0.359.71339 −24.4514 0.3 58.88852 −23.9052 0.3 57.70155 −23.6394 0.356.25029 −23.7127 0.3 54.55685 −23.8064 0.3 52.6178 −23.9101 0.350.43328 −24.0211 0.3 48.00633 −24.1359 0.3 45.34269 −24.2494 0.342.45066 −24.3561 0.3 39.34095 −24.4491 0.3 36.02645 −24.52 0.3 32.52218−24.5588 0.3 28.84516 −24.5566 0.3 25.0142 −24.5043 0.3 21.04984−24.3907 0.3 16.97415 −24.206 0.3 12.81045 −23.9436 0.3 8.583199−23.5964 0.3 4.317678 −23.1593 0.3 0.039585 −22.6304 0.3 −4.22535−22.0106 0.3 −8.45158 −21.3034 0.3 −12.6143 −20.5149 0.3 −16.6897−19.6548 0.3 −20.6556 −18.7354 0.3 −24.4916 −17.7716 0.3 −28.1792−16.7812 0.3 −31.7027 −15.7849 0.3 −35.0507 −14.8106 0.3 −38.2135−13.8867 0.3 −41.1869 −13.0526 0.3 −43.9703 −12.36 0.3 −46.5617 −11.870.3 −48.9505 −11.6638 0.3 −51.0981 −11.801 0.3 −52.9355 −12.2952 0.3−54.4047 −13.0531 0.3 −55.5124 −13.919 0.3 −56.3172 −14.7612 0.3−56.8881 −15.4993 0.3 −57.2838 −16.0942 0.3 −57.5467 −16.5345 0.3−57.9331 −12.9372 0.4 −58.0485 −13.1656 0.4 −58.2071 −13.502 0.4 −58.43−14.0309 0.4 −58.6953 −14.7854 0.4 −58.9576 −15.7964 0.4 −59.147−17.0865 0.4 −59.1723 −18.6601 0.4 −58.9292 −20.4944 0.4 −58.3196−22.5352 0.4 −57.2683 −24.7024 0.4 −55.7336 −26.9041 0.4 −53.7078−29.0514 0.4 −51.2089 −31.067 0.4 −48.2734 −32.8924 0.4 −44.9489−34.4895 0.4 −41.2866 −35.8387 0.4 −37.3365 −36.9324 0.4 −33.146−37.7726 0.4 −28.7591 −38.3682 0.4 −24.2168 −38.7332 0.4 −19.557−38.8844 0.4 −14.8149 −38.8401 0.4 −10.0236 −38.6209 0.4 −5.21395−38.2486 0.4 −0.41512 −37.746 0.4 4.345489 −37.1359 0.4 9.041644−36.4385 0.4 13.64823 −35.6727 0.4 18.14189 −34.859 0.4 22.50055−34.0168 0.4 26.70317 −33.1628 0.4 30.72967 −32.3099 0.4 34.56199−31.4728 0.4 38.18357 −30.6636 0.4 41.57941 −29.8921 0.4 44.73628−29.1658 0.4 47.64333 −28.4916 0.4 50.29196 −27.8743 0.4 52.67616−27.3175 0.4 54.79269 −26.8229 0.4 56.64139 −26.3912 0.4 58.22524−26.0212 0.4 59.4554 −25.4377 0.4 60.18516 −24.6047 0.4 60.54537−23.8147 0.4 60.67636 −23.1795 0.4 60.69517 −22.7272 0.4 60.67531−22.4422 0.4 60.64494 −22.225 0.4 60.60519 −22.0331 0.4 60.53798−21.7868 0.4 60.39301 −21.4093 0.4 60.10812 −20.9048 0.4 59.59779−20.3206 0.4 58.77272 −19.7746 0.4 57.58566 −19.5092 0.4 56.13443−19.5831 0.4 54.44102 −19.6774 0.4 52.50201 −19.7816 0.4 50.31753−19.8935 0.4 47.89062 −20.0091 0.4 45.22701 −20.1236 0.4 42.33503−20.2312 0.4 39.22534 −20.3253 0.4 35.91087 −20.3974 0.4 32.40662−20.4374 0.4 28.72959 −20.4365 0.4 24.89861 −20.3855 0.4 20.93421−20.2733 0.4 16.85846 −20.0901 0.4 12.69467 −19.8291 0.4 8.467298−19.4834 0.4 4.201625 −19.0478 0.4 −0.07665 −18.5203 0.4 −4.3418−17.9021 0.4 −8.56828 −17.1963 0.4 −12.7313 −16.4093 0.4 −16.807−15.5506 0.4 −20.7732 −14.6326 0.4 −24.6095 −13.6701 0.4 −28.2975−12.681 0.4 −31.8214 −11.6859 0.4 −35.1696 −10.7128 0.4 −38.3328 −9.790.4 −41.3065 −8.95694 0.4 −44.0901 −8.26535 0.4 −46.6817 −7.7762 0.4−49.0706 −7.57089 0.4 −51.2181 −7.70876 0.4 −53.0554 −8.20364 0.4−54.5243 −8.96205 0.4 −55.6317 −9.82837 0.4 −56.4362 −10.6708 0.4−57.0068 −11.4091 0.4 −57.4023 −12.0042 0.4 −57.665 −12.4446 0.4−58.0055 −9.16141 0.5 −58.1195 −9.39048 0.5 −58.2761 −9.72777 0.5−58.4961 −10.258 0.5 −58.7569 −11.014 0.5 −59.0135 −12.0265 0.5 −59.1954−13.3176 0.5 −59.2117 −14.8913 0.5 −58.958 −16.7243 0.5 −58.3367−18.7615 0.5 −57.2729 −20.9226 0.5 −55.7255 −23.1154 0.5 −53.6874−25.251 0.5 −51.1769 −27.2522 0.5 −48.231 −29.0607 0.5 −44.8973 −30.63860.5 −41.2273 −31.9666 0.5 −37.271 −33.0376 0.5 −33.0757 −33.8536 0.5−28.6855 −34.424 0.5 −24.1412 −34.7628 0.5 −19.4805 −34.8872 0.5−14.7388 −34.8156 0.5 −9.94879 −34.5687 0.5 −5.14137 −34.1688 0.5−0.34552 −33.6385 0.5 4.411495 −33.001 0.5 9.103557 −32.2766 0.513.70565 −31.4842 0.5 18.19456 −30.6447 0.5 22.54829 −29.7774 0.526.74593 −28.8992 0.5 30.76744 −28.0232 0.5 34.59488 −27.164 0.538.21174 −26.3339 0.5 41.60308 −25.5429 0.5 44.75571 −24.7984 0.547.65882 −24.1074 0.5 50.30386 −23.4749 0.5 52.68482 −22.9044 0.554.79847 −22.3976 0.5 56.64464 −21.9553 0.5 58.22634 −21.5761 0.559.45311 −20.9856 0.5 60.17807 −20.1484 0.5 60.53373 −19.3563 0.560.66105 −18.7204 0.5 60.67726 −18.268 0.5 60.65576 −17.9831 0.560.62414 −17.7661 0.5 60.58328 −17.5744 0.5 60.51466 −17.3286 0.560.3675 −16.9518 0.5 60.07972 −16.449 0.5 59.56603 −15.8677 0.5 58.73783−15.3265 0.5 57.54926 −15.0679 0.5 56.09848 −15.1502 0.5 54.40564−15.2542 0.5 52.46726 −15.3697 0.5 50.28346 −15.4941 0.5 47.85726−15.6237 0.5 45.19436 −15.7535 0.5 42.30304 −15.8778 0.5 39.19395−15.9898 0.5 35.87995 −16.081 0.5 32.37598 −16.1411 0.5 28.69901−16.1615 0.5 24.86781 −16.1325 0.5 20.90282 −16.0431 0.5 16.82609−15.8834 0.5 12.66086 −15.6464 0.5 8.431566 −15.325 0.5 4.163455 −14.9140.5 −0.11779 −14.4112 0.5 −4.38643 −13.8175 0.5 −8.6169 −13.1361 0.5−12.7843 −12.3731 0.5 −16.865 −11.5379 0.5 −20.8364 −10.6427 0.5−24.6782 −9.70237 0.5 −28.3718 −8.73454 0.5 −31.9013 −7.75972 0.5−35.2551 −6.80597 0.5 −38.4236 −5.90136 0.5 −41.402 −5.08545 0.5−44.1896 −4.4099 0.5 −46.7839 −3.93569 0.5 −49.174 −3.74414 0.5 −51.3207−3.89437 0.5 −53.1551 −4.39983 0.5 −54.6195 −5.16669 0.5 −55.722−6.03937 0.5 −56.5216 −6.88645 0.5 −57.088 −7.62804 0.5 −57.48 −8.225350.5 −57.7402 −8.66727 0.5 −58.1585 −4.99126 0.6 −58.2728 −5.22021 0.6−58.4298 −5.55733 0.6 −58.6503 −6.08732 0.6 −58.9119 −6.84304 0.6−59.1695 −7.85526 0.6 −59.3528 −9.1462 0.6 −59.3707 −10.7199 0.6 −59.119−12.5531 0.6 −58.4998 −14.591 0.6 −57.4382 −16.7532 0.6 −55.8932−18.9476 0.6 −53.8573 −21.0854 0.6 −51.3489 −23.0892 0.6 −48.4049−24.9007 0.6 −45.0729 −26.4822 0.6 −41.4043 −27.814 0.6 −37.4491−28.8891 0.6 −33.2546 −29.7096 0.6 −28.865 −30.2845 0.6 −24.3211−30.6281 0.6 −19.6606 −30.7574 0.6 −14.9187 −30.6907 0.6 −10.1285−30.4489 0.6 −5.32066 −30.054 0.6 −0.52425 −29.5288 0.6 4.233432−28.8962 0.6 8.926249 −28.1767 0.6 13.52917 −27.3891 0.6 18.01895−26.5543 0.6 22.3736 −25.6916 0.6 26.57215 −24.8177 0.6 30.59458 −23.9460.6 34.42291 −23.0908 0.6 38.04064 −22.2645 0.6 41.43281 −21.4771 0.644.58622 −20.7359 0.6 47.49005 −20.0479 0.6 50.13575 −19.4182 0.652.5173 −18.8501 0.6 54.63148 −18.3456 0.6 56.47812 −17.9052 0.658.06021 −17.5277 0.6 59.2876 −16.9384 0.6 60.01343 −16.1021 0.660.36992 −15.3103 0.6 60.49791 −14.6746 0.6 60.51459 −14.2221 0.660.49339 −13.9372 0.6 60.462 −13.7201 0.6 60.42134 −13.5284 0.6 60.35297−13.2825 0.6 60.20621 −12.9056 0.6 59.91896 −12.4025 0.6 59.40588−11.8207 0.6 58.57825 −11.2786 0.6 57.38995 −11.0188 0.6 55.93908−11.0995 0.6 54.24613 −11.2018 0.6 52.30764 −11.3152 0.6 50.12371−11.4373 0.6 47.69737 −11.5644 0.6 45.03433 −11.6914 0.6 42.14288−11.8127 0.6 39.03368 −11.9215 0.6 35.71959 −12.0092 0.6 32.21556−12.0656 0.6 28.53857 −12.0821 0.6 24.7074 −12.0491 0.6 20.74251−11.9556 0.6 16.66594 −11.7916 0.6 12.50097 −11.5502 0.6 8.272009−11.2245 0.6 4.004331 −10.8089 0.6 −0.27638 −10.3017 0.6 −4.5444−9.70352 0.6 −8.77416 −9.01767 0.6 −12.9408 −8.25031 0.6 −17.0205−7.41084 0.6 −20.9911 −6.51148 0.6 −24.8318 −5.56711 0.6 −28.5244−4.59542 0.6 −32.053 −3.61691 0.6 −35.4058 −2.65965 0.6 −38.5732−1.75172 0.6 −41.5508 −0.93269 0.6 −44.3377 −0.25422 0.6 −46.93150.222706 0.6 −49.3214 0.416763 0.6 −51.4682 0.268773 0.6 −53.3031−0.23476 0.6 −54.7684 −1.00009 0.6 −55.8718 −1.87162 0.6 −56.6723−2.71786 0.6 −57.2395 −3.45886 0.6 −57.6321 −4.05576 0.6 −57.8928−4.4974 0.6 −58.3082 0.227378 0.7 −58.4269 0.000643 0.7 −58.5902−0.33343 0.7 −58.8208 −0.85913 0.7 −59.0968 −1.60973 0.7 −59.3736−2.61687 0.7 −59.5814 −3.90409 0.7 −59.6292 −5.47719 0.7 −59.4124−7.31484 0.7 −58.8321 −9.36411 0.7 −57.8118 −11.5461 0.7 −56.3088−13.7696 0.7 −54.314 −15.9457 0.7 −51.8442 −17.9968 0.7 −48.9351 −19.8640.7 −45.6338 −21.5086 0.7 −41.9912 −22.91 0.7 −38.0572 −24.0601 0.7−33.8791 −24.9602 0.7 −29.5012 −25.6186 0.7 −24.9646 −26.0485 0.7−20.3074 −26.2664 0.7 −15.5652 −26.29 0.7 −10.7712 −26.1393 0.7 −5.95671−25.8359 0.7 −1.15118 −25.4021 0.7 3.617671 −24.8601 0.7 8.323325 −24.230.7 12.9404 −23.5302 0.7 17.44525 −22.7809 0.7 21.81551 −22.0012 0.726.02992 −21.2073 0.7 30.06821 −20.4122 0.7 33.91212 −19.63 0.7 37.54491−18.8728 0.7 40.95145 −18.15 0.7 44.11839 −17.4689 0.7 47.03478 −16.83630.7 49.69198 −16.257 0.7 52.0839 −15.7344 0.7 54.2073 −15.2702 0.756.06199 −14.865 0.7 57.65098 −14.5176 0.7 58.88935 −13.9518 0.759.63096 −13.1294 0.7 60.00244 −12.3446 0.7 60.14251 −11.7114 0.760.16779 −11.2594 0.7 60.15201 −10.9741 0.7 60.12475 −10.7564 0.760.08775 −10.564 0.7 60.02407 −10.3168 0.7 59.88451 −9.9372 0.7 59.60688−9.42869 0.7 59.10496 −8.83723 0.7 58.28779 −8.2795 0.7 57.10465−7.99713 0.7 55.65251 −8.05024 0.7 53.95792 −8.1203 0.7 52.01761−8.19683 0.7 49.83176 −8.27736 0.7 47.40344 −8.35827 0.7 44.73847−8.43459 0.7 41.84524 −8.50084 0.7 38.73453 −8.55047 0.7 35.41937−8.57509 0.7 31.9149 −8.56489 0.7 28.23827 −8.51143 0.7 24.40841−8.40558 0.7 20.44602 −8.23666 0.7 16.37331 −7.99512 0.7 12.21368−7.67457 0.7 7.991685 −7.26839 0.7 3.732684 −6.77178 0.7 −0.53761−6.18315 0.7 −4.79347 −5.50393 0.7 −9.00942 −4.73774 0.7 −13.1607−3.89126 0.7 −17.2237 −2.97433 0.7 −21.1764 −1.9996 0.7 −24.9985−0.98234 0.7 −28.672 0.059418 0.7 −32.1813 1.104874 0.7 −35.51522.125741 0.7 −38.6649 3.093759 0.7 −41.6264 3.96929 0.7 −44.39984.700645 0.7 −46.9841 5.226829 0.7 −49.3698 5.466313 0.7 −51.51915.35919 0.7 −53.3633 4.890649 0.7 −54.8429 4.153337 0.7 −55.96263.302956 0.7 −56.7791 2.472098 0.7 −57.3602 1.742021 0.7 −57.76421.152699 0.7 −58.0331 0.716093 0.7 −58.4228 3.615096 0.8 −58.53863.386908 0.8 −58.6979 3.050833 0.8 −58.9219 2.522325 0.8 −59.18851.768357 0.8 −59.4528 0.757863 0.8 −59.6447 −0.53184 0.8 −59.673−2.10541 0.8 −59.4334 −3.94022 0.8 −58.8278 −5.98214 0.8 −57.7806−8.15135 0.8 −56.2501 −10.356 0.8 −54.2285 −12.5072 0.8 −51.7334−14.5276 0.8 −48.8014 −16.3586 0.8 −45.48 −17.9621 0.8 −41.8203 −19.31830.8 −37.8724 −20.4196 0.8 −33.6834 −21.2678 0.8 −29.2977 −21.8719 0.8−24.7561 −22.2456 0.8 −20.0966 −22.4058 0.8 −15.3545 −22.3706 0.8−10.5627 −22.1605 0.8 −5.75235 −21.7975 0.8 −0.95257 −21.3041 0.83.809203 −20.7031 0.8 8.506688 −20.0148 0.8 13.11473 −19.2577 0.817.60995 −18.4527 0.8 21.97022 −17.6189 0.8 26.17447 −16.7729 0.830.2026 −15.9278 0.8 34.03652 −15.098 0.8 37.65965 −14.2958 0.8 41.05696−13.5309 0.8 44.21523 −12.8106 0.8 47.12355 −12.1419 0.8 49.77337−11.5298 0.8 52.15864 −10.9775 0.8 54.27611 −10.487 0.8 56.12564−10.0589 0.8 57.7102 −9.69184 0.8 58.94147 −9.11072 0.8 59.67283−8.27919 0.8 60.03455 −7.48983 0.8 60.16677 −6.85494 0.8 60.18644−6.40265 0.8 60.16713 −6.11761 0.8 60.13718 −5.9003 0.8 60.0978 −5.708330.8 60.03106 −5.46197 0.8 59.88681 −5.08412 0.8 59.60289 −4.57908 0.859.09368 −3.99389 0.8 58.26966 −3.44633 0.8 57.08311 −3.17864 0.855.63174 −3.24974 0.8 53.93816 −3.34079 0.8 51.99894 −3.44136 0.849.81426 −3.54897 0.8 47.38713 −3.65997 0.8 44.72331 −3.7693 0.841.83112 −3.8714 0.8 38.72126 −3.95957 0.8 35.40666 −4.02527 0.831.90234 −4.05849 0.8 28.22532 −4.05059 0.8 24.39445 −3.99221 0.820.43027 −3.87241 0.8 16.35488 −3.68136 0.8 12.1916 −3.41237 0.87.964892 −3.05854 0.8 3.700064 −2.61474 0.8 −0.57719 −2.07907 0.8−4.84114 −1.45264 0.8 −9.06626 −0.73875 0.8 −13.2277 0.056221 0.8−17.3018 0.92273 0.8 −21.2663 1.848407 0.8 −25.1007 2.818224 0.8−28.7867 3.814388 0.8 −32.3087 4.816278 0.8 −35.6551 5.795754 0.8−38.8165 6.724667 0.8 −41.7886 7.563434 0.8 −44.5709 8.260365 0.8−47.1615 8.754487 0.8 −49.55 8.96439 0.8 −51.6978 8.830641 0.8 −53.5368.339285 0.8 −55.0064 7.583695 0.8 −56.1155 6.719503 0.8 −56.92165.878592 0.8 −57.4936 5.14137 0.8 −57.8902 4.547088 0.8 −58.15384.107182 0.8 −57.8307 11.92126 0.9 −57.9626 11.70201 0.9 −58.145611.37827 0.9 −58.4071 10.86727 0.9 −58.7274 10.13447 0.9 −59.06389.145632 0.9 −59.348 7.873097 0.9 −59.4896 6.305652 0.9 −59.38284.458349 0.9 −58.9258 2.37811 0.9 −58.0375 0.139118 0.9 −56.6698−2.17004 0.9 −54.8084 −4.46124 0.9 −52.4653 −6.65611 0.9 −49.6728−8.69352 0.9 −46.4755 −10.5321 0.9 −42.923 −12.1483 0.9 −39.0646 −13.5310.9 −34.9476 −14.6788 0.9 −30.6168 −15.5971 0.9 −26.1139 −16.297 0.9−21.478 −16.7923 0.9 −16.7456 −17.0987 0.9 −11.9512 −17.2343 0.9−7.12719 −17.2187 0.9 −2.30433 −17.0722 0.9 2.488356 −16.8157 0.97.223215 −16.4675 0.9 11.87382 −16.0443 0.9 16.41534 −15.5651 0.920.82434 −15.0475 0.9 25.0786 −14.5065 0.9 29.15713 −13.9537 0.933.04085 −13.4022 0.9 36.71235 −12.863 0.9 40.15593 −12.3447 0.943.35786 −11.8538 0.9 46.3068 −11.3962 0.9 48.99382 −10.9765 0.951.41267 −10.5975 0.9 53.55997 −10.2608 0.9 55.43553 −9.96695 0.957.04241 −9.71498 0.9 58.31233 −9.22405 0.9 59.10168 −8.44735 0.959.51932 −7.68609 0.9 59.69691 −7.06237 0.9 59.74911 −6.61268 0.959.75038 −6.32698 0.9 59.73615 −6.10808 0.9 59.7107 −5.91377 0.959.66188 −5.66324 0.9 59.54521 −5.27598 0.9 59.29841 −4.75181 0.958.83266 −4.13147 0.9 58.05022 −3.52598 0.9 56.88603 −3.17354 0.955.43331 −3.13992 0.9 53.73756 −3.10876 0.9 51.79615 −3.06941 0.949.60938 −3.0194 0.9 47.18056 −2.95531 0.9 44.51579 −2.87251 0.941.62375 −2.76605 0.9 38.51562 −2.63002 0.9 35.2049 −2.45683 0.931.70728 −2.23759 0.9 28.04038 −1.96489 0.9 24.22366 −1.63076 0.920.2784 −1.22577 0.9 16.22735 −0.74171 0.9 12.09426 −0.17358 0.9 7.904010.483736 0.9 3.682219 1.233535 0.9 −0.54535 2.07586 0.9 −4.753123.007748 0.9 −8.91585 4.024077 0.9 −13.0092 5.116693 0.9 −17.01046.274366 0.9 −20.8979 7.48316 0.9 −24.6525 8.72661 0.9 −28.2572 9.9856560.9 −31.6979 11.2386 0.9 −34.965 12.45653 0.9 −38.0513 13.61072 0.9−40.9553 14.66136 0.9 −43.6802 15.55686 0.9 −46.2285 16.23627 0.9−48.5956 16.61765 0.9 −50.7475 16.63893 0.9 −52.6163 16.28124 0.9−54.1373 15.6335 0.9 −55.3057 14.85143 0.9 −56.1703 14.07076 0.9 −56.79413.37665 0.9 −57.2324 12.81248 0.9 −57.5269 12.3927 0.9 −58.249813.69252 1 −58.3728 13.46813 1 −58.5426 13.13728 1 −58.7833 12.61615 1−59.0738 11.87104 1 −59.3701 10.86944 1 −59.6027 9.586489 1 −59.6818.014611 1 −59.4999 6.173114 1 −58.9594 4.112986 1 −57.9816 1.911616 1−56.5219 −0.34054 1 −54.5697 −2.55485 1 −52.14 −4.6535 1 −49.2677−6.5767 1 −45.9988 −8.28486 1 −42.3841 −9.75659 1 −38.4731 −10.9827 1−34.3132 −11.9636 1 −29.9489 −12.7066 1 −25.4215 −13.2244 1 −20.7694−13.5324 1 −16.0285 −13.6479 1 −11.2325 −13.5901 1 −6.41302 −13.38 1−1.59999 −13.0393 1 3.178468 −12.5899 1 7.895444 −12.0511 1 12.52521−11.4408 1 17.04373 −10.7789 1 21.42828 −10.084 1 25.65728 −9.37197 129.71021 −8.65524 1 33.56855 −7.94765 1 37.21533 −7.26089 1 40.63523−6.60422 1 43.81477 −5.98465 1 46.74287 −5.40866 1 49.41079 −4.88096 151.81239 −4.40475 1 53.94438 −3.98175 1 55.80656 −3.61258 1 57.40199−3.29605 1 58.65109 −2.75433 1 59.40849 −1.94644 1 59.79511 −1.16897 159.94741 −0.5386 1 59.98145 −0.08716 1 59.9712 0.198349 1 59.948160.416499 1 59.9149 0.60963 1 59.85602 0.857984 1 59.72383 1.240225 159.4561 1.754026 1 58.96574 2.355093 1 58.15953 2.928548 1 56.982073.233783 1 55.52918 3.208817 1 53.83355 3.171602 1 51.89213 3.132675 149.70512 3.0945 1 47.2757 3.060644 1 44.60975 3.035967 1 41.715773.025773 1 38.60469 3.03642 1 35.28967 3.076024 1 31.78606 3.154113 128.11115 3.278786 1 24.28406 3.458806 1 20.32568 3.704449 1 16.258424.024837 1 12.10578 4.425916 1 7.892447 4.913804 1 3.643865 5.49283 1−0.61422 6.164073 1 −4.85613 6.925604 1 −9.05644 7.773324 1 −13.19068.700063 1 −17.2351 9.695527 1 −21.1681 10.74665 1 −24.9698 11.83775 1−28.6224 12.95048 1 −32.1108 14.06372 1 −35.4243 15.14899 1 −38.554616.17784 1 −41.4986 17.11057 1 −44.2574 17.89552 1 −46.831 18.47167 1−49.2116 18.75732 1 −51.3625 18.69185 1 −53.2155 18.25912 1 −54.70917.55061 1 −55.845 16.72208 1 −56.6775 15.9072 1 −57.2726 15.18852 1−57.6879 14.60713 1 −57.9653 14.17582 1 −58.61 16.52977 1.1 −58.728116.30274 1.1 −58.8906 15.96827 1.1 −59.1199 15.44202 1.1 −59.394114.69074 1.1 −59.6684 13.68292 1.1 −59.873 12.3952 1.1 −59.917 10.821991.1 −59.6957 8.984879 1.1 −59.1104 6.937031 1.1 −58.0849 4.757516 1.1−56.5764 2.537742 1.1 −54.5763 0.36654 1.1 −52.1014 −1.6786 1.1 −49.1878−3.53868 1.1 −45.8825 −5.17513 1.1 −42.2365 −6.56766 1.1 −38.2997−7.70817 1.1 −34.1194 −8.59803 1.1 −29.7399 −9.2457 1.1 −25.2023−9.66459 1.1 −20.5446 −9.87109 1.1 −15.8023 −9.88306 1.1 −11.0087−9.72066 1.1 −6.19496 −9.40555 1.1 −1.3905 −8.95993 1.1 3.377007−8.40632 1.1 8.081107 −7.76477 1.1 12.69646 −7.0536 1.1 17.19946−6.29332 1.1 21.56781 −5.50296 1.1 25.78027 −4.69882 1.1 29.8166−3.89384 1.1 33.65859 −3.10225 1.1 37.28952 −2.3361 1.1 40.69427−1.60498 1.1 43.85954 −0.91619 1.1 46.77438 −0.27646 1.1 49.430160.309308 1.1 51.8208 0.837798 1.1 53.94305 1.307208 1.1 55.79674 1.716911.1 57.38488 2.068172 1.1 58.62187 2.637015 1.1 59.36147 3.461224 1.159.73103 4.246948 1.1 59.86954 4.880494 1.1 59.89372 5.332563 1.159.87725 5.617784 1.1 59.84946 5.835378 1.1 59.81199 6.027737 1.159.74771 6.274748 1.1 59.60722 6.654015 1.1 59.32834 7.161851 1.158.82498 7.75208 1.1 58.00645 8.307811 1.1 56.82262 8.587287 1.155.37061 8.530632 1.1 53.6762 8.456436 1.1 51.73609 8.375166 1.149.55044 8.289291 1.1 47.12233 8.202446 1.1 44.45755 8.119617 1.141.56449 8.046294 1.1 38.45391 7.98907 1.1 35.13882 7.956347 1.131.63434 7.957987 1.1 27.95758 8.002463 1.1 24.12748 8.098953 1.120.16468 8.258186 1.1 16.0914 8.489771 1.1 11.931 8.800164 1.1 7.7080249.196022 1.1 3.447822 9.682229 1.1 −0.82389 10.26042 1.1 −5.081410.92924 1.1 −9.29921 11.68513 1.1 −13.4526 12.52146 1.1 −17.517813.42846 1.1 −21.4729 14.39353 1.1 −25.2974 15.40144 1.1 −28.973416.43422 1.1 −32.4853 17.4711 1.1 −35.8217 18.48382 1.1 −38.973719.44414 1.1 −41.9373 20.31243 1.1 −44.7126 21.03701 1.1 −47.298121.55688 1.1 −49.6844 21.79053 1.1 −51.8334 21.67815 1.1 −53.676521.20511 1.1 −55.1542 20.46418 1.1 −56.2719 19.61107 1.1 −57.086318.77822 1.1 −57.6657 18.04672 1.1 −58.0682 17.45642 1.1 −58.336117.01915 1.1 −58.4214 21.3768 1.2 −58.5423 21.15127 1.2 −58.709 20.818871.2 −58.9449 20.29554 1.2 −59.2284 19.54777 1.2 −59.5154 18.54348 1.2−59.7362 17.25842 1.2 −59.8 15.68589 1.2 −59.6018 13.84615 1.2 −59.042311.79111 1.2 −58.0442 9.598875 1.2 −56.5637 7.360321 1.2 −54.59115.164158 1.2 −52.1421 3.088079 1.2 −49.2521 1.191533 1.2 −45.9676−0.48632 1.2 −42.3394 −1.92456 1.2 −38.4172 −3.11444 1.2 −34.2484−4.05677 1.2 −29.8774 −4.75942 1.2 −25.3454 −5.2353 1.2 −20.6907 −5.50031.2 −15.9489 −5.57187 1.2 −11.1536 −5.46972 1.2 −6.33635 −5.21512 1.2−1.52667 −4.82991 1.2 3.247424 −4.33625 1.2 7.959214 −3.75387 1.212.58314 −3.10075 1.2 17.09534 −2.39711 1.2 21.47327 −1.66171 1.225.69551 −0.91056 1.2 29.74164 −0.15637 1.2 33.59327 0.586883 1.237.23354 1.307339 1.2 40.64721 1.995617 1.2 43.82088 2.644576 1.246.74353 3.24763 1.2 49.40646 3.799981 1.2 51.80355 4.298388 1.253.93153 4.741092 1.2 55.79023 5.127469 1.2 57.38265 5.458747 1.258.62669 6.012 1.2 59.37659 6.826851 1.2 59.756 7.607868 1.2 59.902468.239623 1.2 59.93232 8.691353 1.2 59.91943 8.976758 1.2 59.894389.194685 1.2 59.85933 9.387501 1.2 59.79815 9.635298 1.2 59.6624410.0163 1.2 59.38997 10.5276 1.2 58.89407 11.12411 1.2 58.08258 11.690081.2 56.90236 11.98441 1.2 55.44976 11.94601 1.2 53.75455 11.89311 1.251.81356 11.83622 1.2 49.62701 11.77782 1.2 47.198 11.7215 1.2 44.5323911.67216 1.2 41.63864 11.6352 1.2 38.52758 11.61706 1.2 35.21234 11.6261.2 31.70816 11.67168 1.2 28.03225 11.76235 1.2 24.20366 11.90697 1.220.24318 12.11598 1.2 16.17313 12.39873 1.2 12.01696 12.76138 1.27.79929 13.21028 1.2 3.545534 13.74998 1.2 −0.71858 14.3818 1.2 −4.9673515.10407 1.2 −9.17533 15.91289 1.2 −13.3178 16.80135 1.2 −17.371417.75936 1.2 −21.314 18.77406 1.2 −25.1256 19.82995 1.2 −28.788320.90884 1.2 −32.2868 21.98977 1.2 −35.6103 23.04434 1.2 −38.75 24.044191.2 −41.7025 24.94965 1.2 −44.4684 25.70905 1.2 −47.0472 26.26136 1.2−49.4304 26.52498 1.2 −51.5806 26.43962 1.2 −53.4294 25.98977 1.2−54.9164 25.26748 1.2 −56.0447 24.42847 1.2 −56.8695 23.60592 1.2−57.458 22.88177 1.2 −57.8679 22.29656 1.2 −58.1413 21.8627 1.2 −58.309521.57619 1.2

In the above table, the plane Zadim=0 corresponds to the reference planeP0 situated at the base of the profile. This plane P0 is theintersection of the stacking axis AE of the arm 16B with theaxisymmetric surface 12A of the hub, i.e. it passes via the intersectionbetween the axis AE and the surface 12A where it is assembled with thehub 12. The plane Zadim=1 corresponds to the reference plane P1 situatedat the top of the profile. This plane P1 is the intersection between theaxis AE and the axisymmetric surface 14A of the shroud 14, i.e. itpasses via the intersection of the axis AE with the surface 14A where itis assembled with the shroud 14.

By convention, the stacking axis AE of the arm 16A is the axis thatextends in the radial direction Z and is the axis on which the variousX,Y sections of the arm are “stacked” when designing the arm.

As mentioned at the beginning of the present description, theaerodynamic profile of the disclosure is substantially identical to thenominal profile defined in the above table, i.e. it departs from saidnominal profile by very little at most, being defined in particularwithin an envelope of ±1 mm in a direction normal to the surface of thenominal profile, and/or having X,Y coordinates lying within a range of±5% relative to the coordinates X,Y of the nominal profile.

Also, the aerodynamic profile is positioned rotated within ±10°,particularly within ±5°, more particularly within ±3°, with respect tothe stacking axis of the arm. In other words, the stagger angle may varyby ±10°, particularly by ±5°, more particularly by ±3°.

The coordinates of the above table are given starting from a valueZadim=0 in the reference plane P0. The coordinate Zadim isnon-dimensional, i.e. for a point P, situated at a distance D from theplane P0 (where D is measured along the axis Z), the value of Zadim isD/H, where H represents the total height of the profile as measuredbetween the planes P0 and P1. Naturally, in the table above, bymultiplying the coordinate Zadim by the height H, it is possible toobtain the complete coordinates of the arm.

Preferably, the height H of the profile of the arm lies in the range 144mm ±10%. By way of example, the reference plane P0 is at a distance ofabout 391 mm ±10% from the axis A of the turbine.

In the above table, the profile is characterized by fifteen sectionplanes at constant Zadim coordinates, for which the coordinates X and Yare specified. These fifteen section planes include eleven sectionplanes located between Zadim=0 and Zadim=1, and thus arranged in thefluid stream, these section planes being distributed on the arm's heightfrom P0 to P1. The fifteen section planes also include four sectionplanes which are substantially outside the fluid stream (two sectionplanes with Zadim<0, and two section planes with Zadim>1), these foursection planes being provided for ensuring geometric continuity of thestacking, close to the head and the foot, respectively. In each sectionplane at constant coordinate Zadim, the section of the profile is givenby a continuous and smooth curve, interconnecting all of the points(X,Y). In each section plane, the profile is interpolated so as togenerate a uniform profile.

1. An aerodynamic profile for an arm of a structural casing of a turbinehaving a central hub and a shroud, the arm connecting the central huband the shroud, the profile being, when cold and in a non-coated state,substantially identical to a nominal profile determined by the Cartesiancoordinates X,Y,Zadim given in Table 1, in which the coordinate Zadim isthe quotient D/H, where D is the distance of the point underconsideration from a first reference X,Y plane situated at the base ofthe nominal profile, and H is the height of said profile measured fromsaid first reference plane that is the intersection of the stacking axisof the arm and the axisymmetric surface of the hub, out to a secondreference plane that is the intersection of said stacking axis with theaxisymmetric surface of the shroud, the measurements D and H being takenradially relative to the axis of the turbine, while the coordinate X ismeasured in the axial direction of the turbine.
 2. The aerodynamicprofile as claimed in claim 1, wherein said profile is defined within anenvelope of ±1 mm in a direction normal to the surface of the nominalprofile.
 3. The aerodynamic profile as claimed in claim 1, wherein thecoordinates X,Y of said profile lie within a range of ±5% relative tothe coordinates X,Y of the nominal profile.
 4. The aerodynamic profileas claimed in claim 1, wherein the aerodynamic profile is positionedrotated within ±10° with respect to the stacking axis.
 5. Theaerodynamic profile as claimed in claim 1, wherein the aerodynamicprofile is positioned rotated within ±5° with respect to the stackingaxis.
 6. The aerodynamic profile as claimed in claim 1, wherein theaerodynamic profile is positioned rotated within ±3° with respect to thestacking axis.
 7. The aerodynamic profile as claimed in claim 1, whereinthe arm is a main arm belonging to a series of arms connecting thecentral hub and the shroud, said series comprising at least one main armand at least one secondary arm.
 8. The aerodynamic profile as claimed inclaim 1, wherein the structural casing is an exhaust casing of theturbine.
 9. An arm of a structural casing of a turbine, presenting theaerodynamic profile as claimed in claim
 1. 10. A turbine comprising astructural casing, said structural casing comprising a plurality of armspresenting aerodynamic profiles as claimed in claim
 1. 11. A turbine asclaimed in claim 10, wherein the structural casing is an exhaust casing.12. A turbine comprising a structural casing, said structural casingcomprising a central hub and a shroud, and a series of arms connectingthe central hub and the shroud, said series comprising at least one mainarm and at least one secondary arm, wherein the at least one main armpresents an aerodynamic profile as claimed in claim
 1. 13. The turbineas claimed in claim 12, wherein the structural casing is an exhaustcasing.
 14. The turbine as claimed in claim 12, including between 1 and3 main arms that present the aerodynamic profile.
 15. The turbine asclaimed in claim 12, including a single main arm that presents theaerodynamic profile.
 16. The turbine as claimed in claim 12, wherein theseries of arms comprises between 10 and 20 arms comprising between 1 and3 main arms and between 7 and 19 secondary arms.
 17. The turbine asclaimed in claim 12, wherein the series of arms comprises 15 armscomprising 1 main arm and 14 secondary arms.
 18. The turbine as claimedin claim 16, wherein all the main arms present the aerodynamic profile.